Necessity Is The Mother Of Invention – #ILookLikeAnEngineer

Necessity is the mother of invention.   Unfortunately, the “necessity” can be all too easily forgotten as an essential component in education.    I teach what I teach, in part, out of necessity but it is my necessity not that of my students.   I need to teach the curriculum that I teach because it aligns with the standards set forth by the state but it is not a burning necessity for my students no matter how many times I tell them the essential questions and how they will use it in the future.   Knowing something only becomes a burning necessity in the mind of an eleven year old when they see a need to know it so they can do something they want right now.

So how do we create that need to know?   I think we give kids real problems that they really want to solve.   It’s not something that I can do every day, but I try really hard to find time and space to do it every year.   To do this, I  compact lessons and I accelerate where I can.   This year, I managed to squeeze out almost a month at the end of the year to do an engineering project with my students.

Request For Proposal

Students were presented with a Request For Proposal (RFP) from a fake toy company.    The proposal indicated that this fake toy company was seeking to expand market share to include more girls in their customer base for motorized toys.    The toy company wanted those bidding on the contract to conduct market research and build a toy to meet that need.    The toy company indicated that the toy must meet one of three different criteria:  travel 3 m in 3 s, climb 1 m at a 15 degree slope in 2 s, or climb 1 m at a 30 degree slope.

Creating a Team and Conducting Market Research

Students were assigned teams and formed mini-companies that would bid on the RFP.   They created a team name, logo, and slogan.   Then, they conducted customer surveys with both adults and children in the target age range.    They analyzed the data and determined the type of toy the customer was seeking.

Building Technical Knowledge

ChJmkKSUkAEA9Q-During the same time-frame, students built knowledge of how gear trains work.   They began by building gears on a frame and exploring relationships between the rotations of the gears and the number of teeth on the gears (gear ratios, teeth ratios).   Next, they added a motor and wheels so that they could calculate the rate on a 3 m course and measure the rim force on the wheel.   They repeated this process with gear ratios ranging from 1:3 up to 225:1.   As they did this, they were building important skill in construction as well as an understanding of the different kinds of performance they might expect from different kinds of gear ratios.    From there, they measured rim force on the tooth of a gear connected to the motor.    They did so for different sized gears and then learned how to calculate torque.    With this knowledge, they could explain why certain gear ratios would not move and why certain gear ratios would be well-suited to climbing.   At this point, they had built sufficient knowledge to answer the first stages of that burning question of how to build a toy that would meet each of the criteria.

Making a Prototype

Each team began construction of a basic prototype to meet their desired criteria.   This amounted to attaching the motor and the desired gear train along with the wheels on the frame structure.   Students then tested their motorized frame to see if it met the criteria.   Once they had a basic working prototype, they started constructing a body to give the toy the desired aesthetics.    As they constructed the body, they continued to test the toy to make sure the additional weight did not place them out of compliance with the criteria in the RFP.    They repeated tests multiple times and used median values in order to eliminate outlier trials resulting from poor testing technique.

Sealing the Deal – Writing a Written Proposal and Giving an Oral Presentation


When the toy was completed, each team wrote a written report in response to the RFP and prepared an oral presentation.   The final stage of the project required each team to present their toy to a panel of judges representing the fake toy company.   I recruited 3 engineers and a soon-to-be lawyer to represent both the technical and business interests of the company for the panel of judges.   (I am lucky enough to have Sandia National Laboratories nearby and willing to provide this kind of support to encourage excellence in math and science.)   The judges selected a winning team based on the presentation and a demonstration of the toy.   (The winning team members each got a gift card to Cold Stone Creamery).

While this last stage is not “math”, it is very much a part of what engineers do and I wanted my students to appreciate the importance of being able to communicate effectively as an engineer.  Reading, writing, and speaking are just as much essential skills for an engineer as are math and science mastery

Why It Mattered

  • Students got to experience the engineering process, which is so much more powerful than hearing about it.
  • Girls had to learn how to make something and how to make it work.   It’s not that they are any less adept, but many of them are much less experienced.   This results in a certain amount of hesitancy, initially,   Having to make it work pushes them past this hesitancy and they discover just how good they are at it.    Giving girls this experience and confidence is important in leveling the playing field when it comes to engineering.
  • Students used the math that they have learned this year to do something real that mattered to them (finding unit rates, conducting surveys, making data representations, analyzing data to make decisions, finding medians, using equations to calculate torque, measuring radii).
  • Students had to find ways to work together – teams could not shift part way through the month long project.
  • Students who lacked confidence as speakers learned that public speaking is a learned skill and that you get better at it with practice.   (I made each team do a dry run of their presentation in front of their classmates and get feedback the day before the final presentations.  They took the feedback and were so much better the second day.)

Gallery of Toys



Thinking Outside the Box – Working Forward and Working Backwards with Box and Whisker Plots

Box and whisker plots baffle most middle school students, at least initially.   I think they get lost in the language and acronyms associated with the plot long before they ever try to make sense of the representation.    To address that, several years ago, one of my colleagues and I decided to break the instruction into multiple lessons.    We decided to start by simply focusing on the Five Number Summary the first day in order to build background knowledge.    This would give students the chance to become familiar with the vocabulary and to work with the vocabulary before asking them to apply the concepts inherent in the vocabulary to create a data representation.    Then, after students had a solid grasp of the Five Number Summary, we would move on to creating and analyzing box and whisker plots.

Day One – The Five Number Summary

We started with an NCTM lesson, “It’s All In the Cards”.   Each student starts out with a set of twelve number cards (1-12).   They put them in increasing order and then find the median.    Next, they find the median of the lower half of the data and the median of the upper half of the data.   As they do this, I introduce the Lower Quartile (LQ) and Upper Quartile (UQ) vocabulary.    We repeat the process with 11 cards so that they have the opportunity to work with an odd number of data points.   Next, we repeat the process with 10 and then with 9 cards.  (Initially, I use the numbers in cardinal order.   As we progress, I choose random sets of numbers from the cards.)  At this stage, we start to talk about the parts created by the median, the LQ, and the UQ.    How big are they?   What percentage of the data is in this section of the cards?   As students begin to realize that the data is being broken into quarters, I introduce the idea of Q1 and Q3 as alternate names for the LQ and UQ.    I also ask them where Q2 would be.

Its All In The Cards Cards 1 to 12

For homework, I give students a handout in which they need to find the Five Number Summary for three different data sets.    As I reflected on this first day of instruction, I felt that students had a pretty strong grasp of the concepts.   However, I felt like there was enough time and space in the lesson to increase the cognitive demand.   I didn’t want to jump forward to the box and whisker plot because I wanted them to live with the ideas in the lesson for a little bit of time first.   However, I did want more than I had with just the “find the five number summary” that I had in place.   I decided I would like to try adding several problems both to the lesson and to the homework handout in which I ask the students to essentially work backwards.   I would give them a five number summary and then ask them to create a data set to fit it.   I felt this would require students to think in a different way, to recognize multiple solutions, and provide the opportunity for some rich discourse about why a data set would or would not fit and how to alter a data set to make it fit if needed.   Here is the updated version of the file.   Five Number Summary Version 2 for Higher Cog Demand

Day Two – Making and Analyzing Box and Whisker Plots

I began the lesson by having students create a data set by measuring the time needed to find Waldo in a “Where’s Waldo” picture.    To do this, I took apart a Where’s Waldo book and laminated the pages.   I had students work in pairs: one student searched for Waldo while the other tracked the time and then they reversed roles.    Students then used the times to create a class data set for the time to find Waldo. They also created the Five Number Summary for the class data.


I used the Where’s Waldo Five Number Summary as the basis for instruction on how to make a Box and Whisker Plot.   After creating the box and whisker plot, students worked on a foldable to summarize the main ideas and practice the concept.  (I got the initial version of this foldable from my colleague, Laura, who does not have a blog but should.   I took her version and made a few modifications to fit my needs).




First, students had to summarize the main ideas associated with box and whisker plots.    This forced them to crystalize their thinking.   It also provided them with a resource they could use to study for the upcoming test.


Next, they used a data set provided in the foldable to find the five number summary, to find the range and IQR, and to create a box and whisker plot.  This was the “Working Forward” thinking.

Once they could “Work Forward”, the students had a new twist on the task, “Working Backward”.   They were given a box and whisker plot.   From this, they had to find the five number summary, range and IQR, and determine which of several data sets corresponded to the plot.

Students had one additional opportunity to practice working both forward and backward in the foldable.


Box and Whisker Foldable Version 2

Moving forward, I decided to modify the second “Moving Backward” problem.   I am still providing the box and whisker plot and requiring students to find the five number summary, range and IQR.  However, rather than asking students to select the correct data set, I am asking them to create a data set to match the box and whisker plot.   I could control the level of difficulty by specifying the number of items in the data set.

Homework for Day 2 was the Migraine Medicine handout from the NCTM Navigating Through Data Analysis in Grades 6-8

Weaving a Tapestry – Comparing Data Representations Take 2

As a quilter, I am fascinated by the intricate beauty inherent in fabric art.   Of late, I feel like my work must reflect that same kind of intricate weaving found in a beautiful piece of fabric.   I am trying to help my students see the threads linking the different ways that one can represent data.   While I always try to do that, I have been a lot more explicit with the comparing and contrasting of these representations.   First, I created a Tree Map (a Thinking Map that classifies information) for the ways to represent data.   I will write more about that in coming days.   Next, I created a foldable comparing and contrasting line plots, frequency tables, and histograms.   I wrote about that in a recent post (including a free downloadable version of the foldable      Comparing & Contrasting Line Plots Freq Tables Histograms).   Finally, I created a foldable comparing and contrasting bar graphs, circle graphs, and line graphs.

Comparing & Contrasting Graphs

My intent with this foldable was to help the students see the ways that these three graphs are similar and different.   I wanted them to focus on the kinds of data that each one represents (numerical vs. categorical).   I also wanted them to focus on the kind of information that the representation presents (frequency of a data item, percentage or part of a whole, or the change of one variable with respect to another).   I included a space for an example so they would have a visual representation (which they could also compare and contrast with the representations on the previous compare/contrast foldable).   Finally, I wanted them to focus on the specific common error points for each of the graphs.   My goal was to keep the structure of the foldable similar across the columns of the foldable so that students would read the foldable both vertically and horizontally.   I also tried to keep the basic structure similar to that used in the Comparing and Contrasting Line Plots, Frequency Tables, and Histograms so that they could use the two foldables to see additional connections.

What is a Statistical Question?

As I approached the start of a unit on data, I spent a lot of time thinking about how I was going to meet the needs of two very different students.   I didn’t want this unit to just meet the needs of some “average” or “typical” student.   I was pretty sure I could do that.   I was less confident that teaching to that “typical” kid was going to be enough for these two kids.   Naturally, the two kids were diametrically opposite.   Both of the kids are pretty extreme on the conceptual/sequential cognitive style and have been struggling.   Unfortunately, they are extreme in diametrically opposite directions.   One of them is very sequential in his thinking and tends to compartmentalize information and ideas.   The other one is very conceptual in his thinking and has difficulty managing details and sequential processes.

If I am going to reach both of these kids (in addition to everyone else), I am going to have to build a really good “big picture” that shows the connections between the different ideas very clearly and then fill in the details.   My conceptual thinker needs the big picture to give him a framework on which he can hook the ideas.   He is also going to need some instructional supports to hold on to those details.   I’m going to have to find some ways to help him remember the sequential steps in the processes we address.   My sequential thinker has a hard time building that big picture from the details.   He stays within the details so I have to build the big picture and bring out the connections or he is not going to be able to put the whole picture together.

I decided that I wanted to “start at the very beginning” because “that is a very good place to start”.   (My students had to watch The Sound of Music in their band/orchestra class this week when their teacher was absent.   Being 6th graders, naturally, some of them felt the need to enter class singing.)


I also created some resources to help give the big picture and explicitly bring out those connections.   Those are things I will write about in coming days.

I started by giving the students a quick overview of the unit and then moved into the first lesson.



I decided the very beginning was the question of what exactly is a statistical question.     I wanted my approach to the topic to have a lot of support for a particular student with visual processing challenges.   That meant I needed to make sure there was a lot of verbal discussion and clarity brought forth to enable auditory processing.   It also meant that anything I had on the promethean board was also in some form of a handout (it can be hard for kids with visual processing challenges to read information off of a board).

I started the lesson by presenting my students with a set of about ten questions.   I told students that I wanted them to decide which of those questions were statistical.   They needed to use a Think-Pair-Share technique.   I wanted them to come to their own conclusions first (Think).   I wanted them to discuss their thinking out loud with a partner (Share).   This would help my student who needed to process auditorily. It would also de-privatize mathematical thinking.   I would have a better sense of what my students were thinking as I listened to their discussion.   Struggling students would also have access to the thinking of students who were more readily grasping the idea.   Finally, the whole group discussion would provide the opportunity to clarify the concept.   I wanted everyone engaged so I used a Thumbs Up/Thumbs Down formative assessment to structure the discussion.   For each question, students had to “vote” whether the given question was statistical (thumbs up) or not (thumbs down).


Then, we talked about why the question was or was not statistical.   I was intentional in who I selected to speak.   If there was disagreement, I asked a student from each perspective to justify his or her thinking.   I wanted student voice to drive the conversation and was careful not to tell students what makes a question statistical.   I let students bring out the idea that a statistical question has to be something that would create a table or a graph when answered.

I wanted students to refine their understanding, so I had them do a Write-Pair-Share in which they had to explain what makes a question statistical.   This would require each student to refine his or her thinking, give them the chance to express that thinking and hear the thinking of another student, and then for me to ensure that everyone got the full picture in the whole class discussion.   It also would give my student with visual processing challenges the chance to hear the idea explained several times, several ways, in several different voices.

At this point, I was fairly confident that everyone in the room had a working understanding of the concept.   I wanted to step up the cognitive demand a little bit, so I gave them a second set of questions.   Once again, they had to identify whether each question was statistical.   However, this time, they also had to explain why a given question was not statistical and then re-write the question so that it was statistical.   For this activity, I wanted to ensure that each student did their own thinking but also had the chance to have real discourse.   I decided to have them use the Kagan Numbered Heads Together Cooperative Learning Structure.


In this structure, students work individually on the task.   When a student has completed it, he or she stands up (giving them a chance to move a little bit).   When all the members of the group are standing, the students in the group discuss their responses. (I circulate, listening and sometimes asking a question.)   Students in the group are mutually accountable.   That is, one member of the group will speak for the group, but they don’t know which one.   When the groups were all ready, I led a whole class discussion.   I selected students to explain their response for each question based on seat position within the group.   (By the time class was over, every student had to respond to a question).

Based on student responses, it seemed that students had a good working knowledge of what makes a question statistical.   So, once again, I wanted to increase the cognitive demand.   I gave students a set of four different data representations.   I told them that they must write a statistical question that could have been used to create the data representation.   This was trickier than it might sound.   At first, a lot of the questions were questions that could be answered by analyzing the data rather than questions that would generate the data.   After discussing this subtle difference, things proceeded fairly well.   Again, I had students use a Kagan Cooperative Learning Structure to give a structure for the discourse and to ensure mutual accountability.

I followed up the lesson with an Illustrative Maths performance task.

I feel starting the unit “at the very beginning” was the right place to start.   I was happy with how the lesson seemed to be working for both of my distinctly different students. However, it is early days yet and I have only begun to implement some of my ideas on how to build that big picture and the connections between concepts.   I will have to give it more time to see if my ideas about how I can support these students play out the way that I hope.

While I am generally happy with how the lesson went, there are a few things I want to change for next year.     I gave students a handout with all of the questions and prompts for the lesson.   However, it was pretty cumbersome, taking up multiple pages.   Some   students did all of the work on the handout, folded up the handout and taped it into their interactive notebooks.   Some students cut up each question and then taped it into their notebooks to answer.   I felt like the students who taped the whole handout in probably won’t go back and look at the whole thing to study and those that cut each part out just spent too much time cutting.   I decided that I wanted to use the same structure, but refine the way I gave the information to the students.   For the first set of statistical questions (while they are exploring the idea), I created a card sort with the questions.   I will have students cut out the cards and put them into their interactive notebooks in the form of a T table (Statistical/Non-Statistical).


IMG_0997.JPGStatistical Question Card Sort

I then took the next part of the lesson and created a foldable.   I have two versions.   The first version is my standard version.   It requires students to write their explanation for what defines a question as statistical.   The modified version is a cloze activity in which students need only fill in some blanks.

Identifying Statistical Questions Foldable

Identifying Statistical Questions Foldable Modified